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SageMath
E = EllipticCurve("dl1")
E.isogeny_class()
Elliptic curves in class 251328dl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 251328.dl3 | 251328dl1 | \([0, 1, 0, -8563329, 9642347487]\) | \(264918160154242157473/536027170833\) | \(140516306670845952\) | \([2]\) | \(5898240\) | \(2.5413\) | \(\Gamma_0(N)\)-optimal |
| 251328.dl2 | 251328dl2 | \([0, 1, 0, -8655809, 9423336351]\) | \(273594167224805799793/11903648120953281\) | \(3120469933019176894464\) | \([2, 2]\) | \(11796480\) | \(2.8879\) | |
| 251328.dl4 | 251328dl3 | \([0, 1, 0, 4405631, 35381642207]\) | \(36075142039228937567/2083708275110728497\) | \(-546231622070626811117568\) | \([2]\) | \(23592960\) | \(3.2344\) | |
| 251328.dl1 | 251328dl4 | \([0, 1, 0, -23196929, -30550202529]\) | \(5265932508006615127873/1510137598013239041\) | \(395873510493582535163904\) | \([2]\) | \(23592960\) | \(3.2344\) |
Rank
sage: E.rank()
The elliptic curves in class 251328dl have rank \(1\).
Complex multiplication
The elliptic curves in class 251328dl do not have complex multiplication.Modular form 251328.2.a.dl
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
